A362243 -id:A362243 - cp's OEIS frontend

A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).

5, 7, 9, 11, 13, 15, 17, 17, 19, 21, 23, 25, 25, 27, 27, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 41, 41, 41, 41, 43, 45, 45, 47, 47, 49, 49, 51, 51, 51, 53, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 61, 61, 63, 63, 65, 65, 65, 65, 67, 67, 67, 69, 71, 71, 71, 71, 73, 73, 75, 75, 75, 75, 77

Offset: 1

Robin Visser, Apr 25 2023

nonn

Two elliptic curves over a finite field F_q are isogenous if and only if they have the same trace of Frobenius, or equivalently, have the same number of points over F_q.

Thus, by the Hasse bound, a(n) is the number of integers with absolute value bounded by 2*sqrt(prime(n)).

			For n = 1, the a(1) = 5 isogeny classes of elliptic curves are parametrized by the 5 possible values for the trace of Frobenius: -2, -1, 0, 1, 2.
For n = 2, the a(2) = 7 isogeny classes of elliptic curves are parametrized by the 7 possible values for the trace of Frobenius: -3, -2, -1, 0, 1, 2, 3.

Robin Visser, Table of n, a(n) for n = 1..10000
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
J. H. Silverman, The Arithmetic of Elliptic Curves, Second edition. Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009.

Cf. A247485, A362198, A362201, A362243, A364681.

[2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];

2Floor[2Sqrt[Prime[Range[100]]]]+1 (* Paolo Xausa, Oct 23 2023 *)

PARI
```
a(n) = 2*sqrtint(4*prime(n)) + 1;
```

a(n) = 2*floor(2*sqrt(prime(n))) + 1.

a(n) = 2*A247485(n) - 1.

A363840 a(n) is the number of isomorphism classes of genus 2 hyperelliptic curves over the finite field of order prime(n).

Original entry on oeis.org

20, 69, 285, 749, 2813, 4589, 10149, 14119, 24907, 49675, 60613, 102749, 139613, 160951, 209947, 300667, 414357, 457813, 606151, 721011, 783511, 992477, 1150629, 1418037, 1834951, 2071011, 2196269, 2461749, 2602157, 2898789, 4113149, 4513613, 5161749, 5390839, 6638395, 6909013, 7764749

Offset: 1

Robin Visser, Jun 23 2023

nonn

Any (smooth, projective, geometrically irreducible) curve of genus 2 can be given by a Weierstrass equation of the form: y^2 + h(x)y = f(x), where h(x) and f(x) are polynomials satisfying deg(h) <= 3 and deg(f) <= 6.

			For n = 1, the a(1) = 20 genus 2 curves over F_2 can be given by their Weierstrass models as: y^2 + y = x^5, y^2 + (x^2 + 1)y = x^5, y^2 + (x^3 + x^2 + 1)y = x^5, y^2 + (x^3 + x + 1)y = x^5, y^2 + (x^3 + x^2 + x + 1)y = x^5, y^2 + y = x^5 + x^4, y^2 + (x+1)y = x^5 + x^4, y^2 + (x^2 + x + 1)y = x^5 + x^4, y^2 + (x^3 + x^2 + 1)y = x^5 + x^4, y^2 + y = x^5 + x^4 + x^3, y^2 + (x^2 + x + 1)y = x^5 + x^4 + x^3, y^2 + xy = x^5 + x^4 + x, y^2 + (x^2)y = x^5 + x^4 + x, y^2 + (x^2 + x)y = x^5 + x^4 + x, y^2 + y = x^5 + x^4 + 1, y^2 + (x^2 + x + 1)y = x^5 + x^4 + 1, y^2 + (x^3 + x^2 + 1)y = x^5 + x^4 + 1, y^2 + y = x^5 + x^3 + 1, y^2 + (x^3 + x^2 + 1)y = x^5 + x^3 + 1, and y^2 + (x^3 + x^2 + 1)y = x^6 + x^5 + 1.

G. Cardona, On the number of curves of genus 2 over a finite field, Finite Fields Appl. 9 (2003), no.4, 505-526.
G. Cardona, E. Nart, and J. Pujolàs, Curves of genus two over fields of even characteristic, Math. Z. 250 (2005), no.1, 177-201.
E. Nart, Counting hyperelliptic curves, Adv. Math. 221 (2009), no.3, 774-787.
R. Steinberg, Enumerating Curves of Genus 2 Over Finite Fields, (2018). UVM Honors College Senior Theses. 259.

Cf. A362243, A363843.

def a(n):
    if n == 1: return 20
    p = Primes()[n-1]
    ans = 2*p^3 + p^2 + 2*p - 2
    if p%3 == 1: ans += 2
    if p%5 == 1: ans += 8
    if p == 5: ans += 2
    if p%8 in [1,3]: ans += 2
    return ans

a(1) = 20, and for n > 1, a(n) = 2*prime(n)^3 + prime(n)^2 + 2*prime(n) - 2 + 2*[prime(n) == 1 (mod 3)] + 8*[prime(n) == 1 (mod 5)] + 2*[prime(n) == 5] + 2*[prime(n) == 1 or 3 (mod 8)].

A363843 a(n) is the number of isomorphism classes of genus 3 hyperelliptic curves over the finite field of order prime(n).

Original entry on oeis.org

76, 526, 6508, 34228, 324562, 747004, 2849576, 4965266, 12896050, 41071144, 57316082, 138789292, 231850328, 294172382, 458893426, 836688844, 1430252626, 1689646684, 2700843026, 3609164734, 4146921368, 6155086706, 7879211410, 11169529016, 17176506056, 21022261804, 23187646130

Offset: 1

Robin Visser, Jun 23 2023

nonn

			For n = 1, E. Nart and D. Sadornil showed that there are 76 genus 3 hyperelliptic curves over F_2, so a(1) = 76.

E. Nart, Counting hyperelliptic curves, Adv. Math. 221 (2009), no. 3, 774-787.
E. Nart and D. Sadornil, Hyperelliptic curves of genus three over finite fields of even characteristic, Finite Fields Appl. 10 (2004), no. 2, 198-220.

Cf. A362243, A363840.

def a(n):
    if n == 1: return 76
    p = Primes()[n-1]
    ans = 2*p^5 + 2*p^3 - 2
    if p%4 == 3: ans -= 2*(p^2 - p)
    if p > 3: ans += 2*(p - 1)
    if p%8 == 1: ans += 4
    if p%7 == 1: ans += 12
    if p == 7: ans += 2
    if p%12 in [1, 5]: ans += 2
    return ans

a(1) = 76, and for n > 1, a(n) = 2*prime(n)^5 + 2*prime(n)^3 - 2 - 2*(prime(n)^2 - prime(n))*[prime(n) == 3 (mod 4)] + 2*(prime(n)-1)*[prime(n) > 3] + 4*[prime(n) == 1 (mod 8)] + 12*[prime(n) == 1 (mod 7)] + 2*[prime(n) == 7] + 2*[prime(n) == 1 or 5 (mod 12)].

A369902 Number of isomorphism classes of elliptic curves over the finite field of order prime(n) whose trace of Frobenius is zero.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 4, 6, 6, 6, 2, 8, 4, 10, 6, 12, 6, 4, 14, 4, 10, 12, 12, 4, 14, 10, 12, 6, 8, 10, 20, 8, 12, 14, 14, 6, 4, 22, 14, 20, 10, 26, 4, 10, 18, 12, 14, 20, 10, 12, 30, 12, 28, 16, 26, 22, 22, 6, 20, 12, 18, 12, 38, 8, 10, 12, 8, 20, 14, 16, 38, 18, 10, 12, 34, 22, 6, 20, 16

Offset: 1

Robin Visser, Feb 05 2024

nonn

a(n) is the number of isomorphism classes of elliptic curves E over the finite field F_p such that E has exactly p+1 points over F_p.

			For n = 1, the unique a(1) = 1 elliptic curve over F_2 whose trace of Frobenius is zero is y^2 + y = x^3.
For n = 2, the a(2) = 2 elliptic curves over F_3 whose trace of Frobenius is zero are y^2 = x^3 + x and y^2 = x^3 + 2*x.
For n = 3, the a(3) = 2 elliptic curves over F_5 whose trace of Frobenius is zero are y^2 = x^3 + 1 and y^2 = x^3 + 2.

Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), no. 2, 183-211.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Cf. A259825, A362243.

a(n) = if (n<=2, n, qfbhclassno(4*prime(n)));

# A brute force computation of a(n)
def a(n):
    if n==1: return 1
    p, ECs = Primes()[n-1], []
    for A,B in ((x, y) for x in range(p) for y in range(p)):
        if ((4*A^3 + 27*B^2)%p != 0):
            E = EllipticCurve(GF(p), [A,B])
            if (E.trace_of_frobenius()==0):
                if not any([E.is_isomorphic(Ei) for Ei in ECs]): ECs.append(E)
    return len(ECs)

a(n) = A259825(4*prime(n))/12 if n > 2.

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A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).

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A363840 a(n) is the number of isomorphism classes of genus 2 hyperelliptic curves over the finite field of order prime(n).

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A363843 a(n) is the number of isomorphism classes of genus 3 hyperelliptic curves over the finite field of order prime(n).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Sage

Formula

A369902 Number of isomorphism classes of elliptic curves over the finite field of order prime(n) whose trace of Frobenius is zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Sage

Formula